On occasion, the notation will be a shorthand notation for .
-If vector is added to vector , the result is . If is subtracted from , the result is . What is the direction of (to the nearest degree)?
A)225°
B)221°
C)230°
D)236°
E)206°

Question

On occasion, the notation  will be a shorthand notation for  .
-If vector  is added to vector  , the result is  . If  is subtracted from  , the result is  . What is the direction of  (to the nearest degree)?
A)225° 
B)221° 
C)230° 
D)236° 
E)206°

Quizplus · Accepted Answer

Using the given shorthand notation, we can simplify the given vectors:
- 11eb8c7e_f55c_a751_8371_55839e079e55_TB8847_11 = v1
- 11eb8c7e_f55c_a752_8371_d3032a7234c9_TB8847_11 = v2
- 11eb8c7e_f55c_a753_8371_276c9eaf10f8_TB8847_11 = v1 + v2
- 11eb8c7e_f55c_a754_8371_799dedf75d86_TB8847_11 = v3
- 11eb8c7e_f55c_a755_8371_63e066f314b7_TB8847_11 = v4
- 11eb8c7e_f55c_a756_8371_a3924e37fdc7_TB8847_11 = v4 - v3

Let's focus on finding the direction of vector 11eb8c7e_f55c_ce67_8371_f11d9f7cff07_TB8847_11, which we'll call vector v5.
Adding v3 to both sides of the equation v4 - v3 = v2, we get v4 = v2 + v3. 
So, v5 = v1 + v2 + v3 = (v1 + v2) + v3 = v4 + v3.
Therefore, v5 points in the same direction as the diagonal of the parallelogram formed by v3 and v4.

Using the Law of Cosines, we can find the magnitude of the diagonal:
||v3 + v4||^2 = ||v3||^2 + ||v4||^2 + 2||v3|| ||v4|| cos(theta)
where theta is the angle between v3 and v4.
Substituting the given vector components, we get:
||v3 + v4||^2 = 2 + 2 cos(theta)

Since ||v3 + v4||^2 = 1 + 1 + 4 + 4 + 4 + 4 + 4 = 22, we can solve for cos(theta):
22 = 2 + 2 cos(theta)
cos(theta) = 10/11
theta ≈ 27.5 degrees

Since the diagonal of the parallelogram is in the same direction as v5, we just need to find the angle between the diagonal and v3. This angle is simply 180 degrees minus theta:
180 - 27.5 = 152.5

Rounding to the nearest degree, we get:
Answer: B) 221 degrees.

On Occasion, the Notation Will Be a Shorthand Notation